gmm (version 1.3-0)

getLamb: Solving for the Lagrange multipliers of Generalized Empirical Likelihood (GEL)

Description

It computes the vector of Lagrange multipliers, which maximizes the GEL objective function, using an iterative Newton method.

Usage

getLamb(g, tet, x, type = c('EL','ET','CUE'), tol_lam = 1e-12, maxiterlam = 1000, tol_obj = 1e-7, k = 1)

Arguments

g

A function of the form $g(\theta,x)$ and which returns a $n \times q$ matrix with typical element $g_i(\theta,x_t)$ for $i=1,...q$ and $t=1,...,n$. This matrix is then used to build the q sample moment conditions.

tet

A $k \times 1$ vector of parameters at which the function $g(\theta,x)$ has to be evaluated

x

The matrix or vector of data from which the function $g(\theta,x)$ is computed.

type

"EL" for empirical likelihood, "ET" for exponential tilting and "CUE" for continuous updated estimator.

tol_lam

Tolerance for $\lambda$ between two iterations. The algorithm stops when $\|\lambda_i -\lambda_{i-1}\|$ reaches tol_lam

maxiterlam

The algorithm stops if there is no convergence after "maxiterlam" iterations.

tol_obj

Tolerance for the gradiant of the objective function. The algorithm returns a non-convergence message if $\max(|gradiant|)$ does not reach tol_obj. It helps the gel algorithm to select the right space to look for $\theta$

k

It represents the ratio k1/k2, where $k1=\int_{-\infty}^{\infty} k(s)ds$ and $k2=\int_{-\infty}^{\infty} k(s)^2 ds$. See Smith(2004).

Value

• lambda: A $q\times 1$ vector of Lagrange multipliers which solve the system of equations given above. singular: 0 for a normal solution, 1 if the algorithm does not converge and 2 if the algorithm produces a singular system, NaN or Inf values. conv_mes: A message with details about the convergence.

Details

It solves the problem $\frac{1}{n}\sum_{t=1}^n D\rho()g(\theta,x_t)=0$.

References

Newey, W.K. and Smith, R.J. (2004), Higher Order Properties of GMM and Generalized Empirical Likelihood Estimators. Econometrica, 72, 219-255.

Smith, R.J. (2004), GEL Criteria for Moment Condition Models. Working paper, CEMMAP.

Examples

g <- function(tet,x)
	{
	n <- nrow(x)
	u <- (x[7:n] - tet[1] - tet[2]*x[6:(n-1)] - tet[3]*x[5:(n-2)])
	f <- cbind(u, u*x[4:(n-3)], u*x[3:(n-4)], u*x[2:(n-5)], u*x[1:(n-6)])
	return(f)
	}
n = 500
phi<-c(.2, .7)
thet <- 0.2
sd <- .2
x <- matrix(arima.sim(n = n, list(order = c(2, 0, 1), ar = phi, ma = thet, sd = sd)), ncol = 1)
getLamb(g, c(0, phi), x, type = "EL")